If a TLS certificate public-key algorithm is
ECC secp384r1 or
ECC prime256v1, is it possible to have RSA as a signature algorithm? Or does it use ECDSA for signature? Or can it use any of them (ECDSA or RSA)?
No, RSA cannot be used as a signature algorithm if the certificate has an EC public key in it - you're restricted to ECDSA. You would not be able to verify the signature using the public key after all. The algorithm used must comply with the public key algorithm. However, some keys can be used for different related signature schemes, e.g. RSASSA-PKCS1-v1_5 and RSASSA-PSS (although TLS 1.3 only allows PSS style signatures).
Previously RSA could also be used for session key establishment / authentication using encryption / decryption of the master secret value: the cipher suites starting with
RSA_. That use has been deprecated in TLS 1.3, but it is certainly incompatible with EC certificates in versions earlier than TLS 1.2.